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where a is the distance of the center of mass from the axis and o> the angular velocity of the body. Making these substitutions in equations (2) and (3) and solving for P and Q we obtain
P = - Fn + maw2,                           (VII)
Q = - Ft + mau.                        (VIII)
The magnitude and the direction of R are given by the relations
R =
where < is the angle R makes with the line Oc.
A uniform rod, which is free 'to rotate about a horizontal axis through one end, falls from a horizontal position. Find the reaction of the axis at any instant of- its fall.
Evidently                           Fn=  mg cos 6.
FT   mg sin 9.
The negative sign in the first equation is due to the fact that in equa-* tion (VII) Fn is supposed to be directed towards the axis, while mg cos 6 is directed away from the axis. The negative sign in the second equation is due to the fact that d is measured in the counter-clockwise direction, while mg sin 6 points in the opposite direction.
Substituting these values of Fn and FT in equations (VII) and (VIII),
we obtain
P = mg cos 6 + mao>2,
Q = mg gin 0 + maco.
But by the conservation of energy
i Jco2 = mga cos 0,
where a is one-half the length of the rod.   Therefore
2a and                                     co =